Effect of small variations of parameters in the turboprop cycle

E F F E C T O F SMALL VARIATIONS O F PARAMETERS I N T H E T U R B O P R O P C Y C L E

T h e s i s by

L C d r

.

C a r l O

.

Holrnquist, USN

In P a r t i a l F u l f i l l m e n t of t h e R e q u i r e m e n t s F o r the D e g r e e of

A e r o n a u t i c a l E n g i n e e r

C a l i f o r n i a Institute of Technology P a s a d e n a , C a l i f o r n i a

ACKNOWLEDGEMENT

The author is indebted t o D r . F r a n k E . M a r b l e , Aeronautics D e p a r t m e n t , California Institute of Technology, f o r h i s d i r e c t i o n and

P A R T I I1 I11 I V V

VI

VII

T A B L E O F CONTENTS T I T L E

S u m m a r y I n t r o d u c t i o n

A n a l y s i s of B a s i c T u r b o p r o p C y c l e A n a l y s i s of C y c l e with R e g e n e r a t i o n D i s c u s s i o n of R e s u l t s

C o n c l u s i o n s

S c h e m a t i c D i a g r a m s of T u r b o p r o p C y c l e Appendix A . S y m b o l s a n d Definitions G r a p h s of C o e f f i c i e n t s of E q u a t i o n s

(769,

(77) and (78)

R e f e r e n c e s

SUMMARY

A cycle a n a l y s i s of the t u r b i n e - p r o p e l l e r engine i s given

in t e r m s of the p a r a m e t e r s of the cycle and the component ef-

f i c i e n c i e s of the engine with and without r e g e n e r a t i o n , By m e a n s

of a T a y l o r expansion about the point of i d e a l e f f i c i e n c i e s , t o t a l

power output, s p e c i f i c f u e l c o n s u m p t i o n , and optimum jet p r e s -

s u r e r a t i o a r e given in t e r m s of the i d e a l values plus c o r r e c t i o n s

0

f o r s m a l l v a r i a t i o n s in the component e f f i c i e n c i e s f r o m 100 /o

.

In t h i s way, the r e l a t i v e i m p o r t a n c e of component e f f i c i e n c i e s in affecting the p e r f o r m a n c e of the i d e a l t u r b o p r o p cycle i s demon-

s t r a t e d by m e a n s of s i m p l e a n a l y t i c a l e x p r e s s i o n s involving the

b a s i c cycle p a r a m e t e r s and the component e f f i c i e n c i e s .

The a n a l y s i s of the i d e a l t u r b o p r o p cycle i s given in t e r m s

of t h r e e b a s i c p a r a m e t e r s which a r e functions of the f o r w a r d

flight s p e e d , the c o m p r e s s o r p r e s s u r e r a t i o , and the limiting

combustion c h a m b e r t e m p e r a t u r e . F o r the i d e a l cycle

,

the jet

p r e s s u r e r a t i o f o r optimum division of power between p r o p e l l e r

and jet r e s u l t s in the t o t a l work of the cycle being done by the

p r o p e l l e r

.

Of the component e f f i c i e n c i e s , the t u r b i n e efficiency was

found t o be the m o s t i m p o r t a n t in affecting the p e r f o r m a n c e c r i -

t e r i a of the i d e a l t u r b o p r o p c y c l e . Since a l l w o r k is done by

the p r o p e l l e r in the i d e a l c y c l e , the p r o p e l l e r efficiency was a l s o

a s considered in this a n a l y s i s , d e c r e a s e d the t o t a l power output

and d e c r e a s e d the specific f u e l consumption of the ideal turbine-

-I-

INTRODUCTION

In the a n a l y s i s and d e s i g n of a n a i r c r a f t g a s - t u r b i n e p r o -

p e l l e r - d r i v i n g power p l a n t , it i s i m p o r t a n t not only t o have a

s i m p l i f i e d method of analyzing the cycle with a convenient notation,

but a l s o t o find the r e l a t i v e effect on p e r f o r m a n c e of s m a l l v a r i a -

t i o n s of the component efficiencies in the i d e a l c y c l e . In the a c t u a l

d e s i g n work of component p a r t s of a t u r b o p r o p e n g i n e , the e n g i n e e r

m u s t work with complex p e r f o r m a n c e and c h a r a c t e r i s t i c s c u r v e s

in o r d e r t o obtain quantitative data f o r t h e c o n s t r u c t i o n of the

m a c h i n e . However ,

it

would s e e m t h a t a s i m p l e a n a l y t i c a l ex-

p r e s s i o n , containing the known p a r a m e t e r s of the cycle and giving

the r e l a t i v e effect of a s m a l l change in component efficiencies on

the p e r f o r m a n c e of the i d e a l cycle , would be helpful in o v e r a l l

d e s i g n c o n s i d e r a t i o n s .

The p r o b l e m of analyzing the flow in the g a s - t u r b i n e power

plant h a s been t r e a t e d e x t e n s i v e l y ( s e e R e f e r e n c e s ) . A method

f o r d e t e r m i n i n g the optimum division of power between jet and

p r o p e l l e r f o r m a x i m u m t h r u s t h a s b e e n g r a p h i c a l l y analyzed in

R e f e r e n c e 3 . T h i s method involves the use of complex c u r v e s in

o r d e r t o find the optimum jet p r e s s u r e r a t i o f o r m a x i m u m t h r u s t

with a given s e t of engine o p e r a t i n g conditions. It would s e e m

l e s s c u m b e r s o m e if a s i m p l e a n a l y t i c a l e x p r e s s i o n could be de-

-

2-

v a r i a t i o n t h e r e f r o m f o r a s m a l l change in component e f f i c i e n c i e s .

An a n a l y s i s of the b a s i c t u r b o p r o p c y c l e , with and without

r e g e n e r a t i o n , was m a d e i n t e r m s of the p a r a m e t e r s of the s y s t e m .

In t h i s a n a l y s i s , the weight of f u e l injected in the combustion

c h a m b e r and the v a r i a t i o n of s p e c i f i c h e a t s w e r e n e g l e c t e d , Com-

bustion efficiency was a s s u m e d t o be 100O/~, and the m o m e n t u m

p r e s s u r e l o s s in the combustion c h a m b e r was a l s o neglected. Isen-

t r o p i c , f r i c t i o n l e s s c o m p r e s s i o n i n the inlet d i f f u s e r was a s s u m e d .

F r o m the cycle a n a l y s i s , p e r f o r m a n c e e x p r e s s i o n s w e r e d e r i v e d

f o r t o t a l power output and s p e c i f i c f u e l consumption. A r e l a t i o n

defining the optimum jet p r e s s u r e r a t i o was obtained f r o m the

t o t a l power output of the c y c l e .

The r e l a t i v e effect on p e r f o r m a n c e c r i t e r i a of s m a l l

changes in component e f f i c i e n c i e s was t h e n investigated t o s h o w ,

individually, the influence of 10s s e s i n e a c h component. Using the r e l a t i o n s d e r i v e d f o r t o t a l power o u t p u t , s p e c i f i c f u e l consump-

t i o n , and optimum jet p r e s s u r e r a t i o , a T a y l o r e x p a n s i o n was used

0

f r o m t h e i d e a l o p e r a t i n g point ( a l l component e f f i c i e n c i e s 100 /o)

.

The coefficients of t h e s e d e r i v e d e x p a n s i o n s w e r e then used t o

show the r e l a t i v e effect of component e f f i c i e n c i e s on the i d e a l p e r -

I. ANALYSIS O F THE BASIC TURBOPROP CYCLE

A s chernatic d i a g r a m of the b a s i c t u r b i n e - p r o p e l l e r engine without r e g e n e r a t i o n i s p r e s e n t e d a s F i g u r e 1. The various points

in the cycle through which the working fluid p a s s e s a r e n u m b e r e d

and a r e r e f e r r e d t o in the d e r i v a t i o n , Symbols used in the analy-

s i s a r e defined in Appendix A . E x p r e s s i o n s f o r work output of component p a r t s , t e m p e r a t u r e r a t i o s a t v a r i o u s points in the i d e a l

and non-ideal cycle

,

and a l l p e r f o r m a n c e c r i t e r i a have been de-

r i v e d in t e r m s of f o u r b a s i c p a r a m e t e r s of the cycle:

The f a c t o r /

,

t h e r e f o r e , i s r e l a t e d t o the a i r c r a f t flight s p e e d .

Th- f a c t o r

b

i s r e l a t e d t o the p r e s s u r e r a t i o a c r o s s the com-

p r e s s o r .

'The f a c t o r K i s r e l a t e d t o the t e m p e r a t u r e r i s e in the combustion c h a m b e r a n d , h e n c e , t o the quantity of f u e l injected and b u r n e d .

When the f a c t o r s p

,

6 ,

and K have been p r e s c r i b e d in any given t u r b o p r o p c y c l e , t h e r e i s one additional p a r a m e t e r needed t o de-

a s the jet p r e s s u r e r a t i o X , and defines the p r e s s u r e and t e m p e r - a t u r e r a t i o a c r o s s the jet e x h a u s t n o z z l e . The value of the jet

p r e s s u r e r a t i o in any given cycle d e t e r m i n e s the p r o p o r t i o n of

the t o t a l work of the engine t h a t goes t o d r i v i n g the p r o p e l l e r and

the p r o p o r t i o n t h a t goes t o jet t h r u s t . If the optimum jet p r e s s u r e

r a t i o

x':

f o r optimum division of power between p r o p e l l e r and j e t ,

i s introduced in t e r m s o f p ,

6,

and K

,

t h e n the p e r f o r m a n c e c r i - t e r i a of the complete i d e a l cycle c a n be e x p r e s s e d i n t e r m s of

t h e s e t h r e e p a r a m e t e r s only.

The B a s i c T u r b o p r o p e l l e r Cycle. In the b a s i c c y c l e , a i r

p a s s e s f r o m f r e e s t r e a m a t s t a t i o n 0 and i s c o m p r e s s e d i s e n -

t r o p i c a l l y and without f r i c t i o n in the inlet diffuser t o s t a t i o n 1.

Stagnation s t a t e conditions a r e c o n s i d e r e d a t the c o m p r e s s o r in-

l e t :

C o m p r e s s i o n in the c o m p r e s s o r i s c a r r i e d out between

s t a t i o n s 1 and 2 , Introducing the notation d e s c r i b e d a b o v e , the

The work of the c o m p r e s s o r f o r i d e a l and non-ideal c o m p r e s s i o n i s :

The heat added a t s t a t i o n 3 in the dombustion c h a m b e r p e r pound

of a i r i s given below. Combustion e f f i c i e n c y i s a s s u m e d

t o

be 100O/o

and the m o m e n t u m p r e s s u r e l o s s in the combustion c h a m b e r i s

n e g l e c t e d .

Qc

7-3

=

C ~ & - T ; ) =

c P z ( =

-+)

K =

G

_To

The t u r b i n e in the t u r b o p r o p c y c l e not only s u p p l i e s the

power t o d r i v e the c o m p r e s s o r , but a l s o d r i v e s the p r o p e l l e r .

A hypothetical point 4 of the t u r b i n e h a s b e e n c h o s e n t o t h e o r e t - i c a l l y divide the t u r b i n e w o r k between c o m p r e s s o r and propel-

l e r . By equating the w o r k of the c o m p r e s s o r t o the work of the

t u r b i n e t o s t a t i o n 4 , the following t e m p e r a t u r e r a t i o s a r e derived:

Having obtained a n e x p r e s s i o n f o r the t e m p e r a t u r e r a t i o a t point 4 , the work t o d r i v e the p r o p e l l e r c a n be d e r i v e d by introducing the

Knowing t h e s e t e m p e r a t u r e r a t i o s in t e r m s of the cycle p a r a m e t e r s ,

the jet velocity

V6

c a n be d e t e r m i n e d :

The output of the t u r b o p r o p cycle c a n be e x p r e s s e d e i t h e r in pounds

t h r u s t o r i n power d i m e n s i o n s . Since it i s believed t h a t the output

of the g a s - t u r b i n e p r o p e l l e r - d r i v i n g engine c a n b e s t be given in

t e r m s of p o w e r , t h e s e d i m e n s i o n s a r e u s e d throughout the a n a l y s i s .

The t o t a l output of the cycle without r e g e n e r a t i o n including

p r o p e l l e r work and work t o t h r u s t is:

Optimum J e t P r e s s u r e R a t i o . Equation

(19)

gives the t o t a l

power output of the t u r b o p r o p cycle in t e r m s of f o u r b a s i c p a r a m -

e t e r s p

,

6,

K , and X and the component e f f i c i e n c i e s . By i n t r o - ducing the optimum jet p r e s s u r e r a t i o X* f o r i d e a l division of power between jet and p r o p e l l e r , the b a s i c p a r a m e t e r s f o r the i d e a l cycle

c a n be r e d u c e d t o t h r e e , n a m e l y ,

b,p,

and K . In o r d e r t o obtain a n e x p r e s s i o n defining the optimum jet p r e s s u r e r a t i o X:@, the t o t a l work r e l a t i o n i s d i f f e r e n t i a t e d with r e s p e c t t o X and s e t e q u a l t o z e r o . The r e l a t i o n defining the optimum jet p r e s s u r e r a t i o X:: i s t h e n found t o be:

F o r i d e a l e f f i c i e n c i e s

(qp=

tp=

CY

=

)

t h i s r e d u c e s

It i s i n t e r e s t i n g t o note t h a t the t o t a l w o r k e x p r e s s i o n f o r the t u r b o -

p r o p cycle , with a l l component e f f i c i e n c i e s 1 0 0 ~ / o and with optimum

jet p r e s s u r e r a t i o , r e d u c e s t o :

and i s obtained with s u c h a jet p r e s s u r e r a t i o X:: t h a t the jet velocity

V6

i s e q u a l t o the f r e e s t r e a m velocity VO, T h u s f o r the i d e a l t u r b o -

p r o p c y c l e , the t o t a l work output would go t o the p r o p e l l e r if the

condition of optimum jet p r e s s u r e r a t i o is i m p o s e d , T h i s follows

f r o m the definition of o v e r a l l efficiency of the t u r b o j e t and p r o p e l l e r

c y c l e s . F r o m kinetic e n e r g y c o n s i d e r a t i o n s , the following e x p r e s -

s i o n f o r o v e r a l l efficiency of jet and p r o p e l l e r d r i v e n s y s t e m s c a n

be e a s i l y d e r i v e d :

2

I

=

l+~

Vb ( 2 3 )

H e r e V r e p r e s e n t s the jet velocity o r the velocity behind the pro- p e l l e r d i s c , and V r e p r e s e n t s f r e e s t r e a m velocity. T h u s , maxi-

0

m u m efficiency i s obtained f o r a jet cycle when the velocity of the

jet i s e q u a l t o the f r e e s t r e a m velocity. The condition imposed on

the i d e a l t u r b o p r o p cycle that p r o p e l l e r efficiency 51P i s 10o0/0,

-

10

-

change in velocity a c r o s s the d i s c , Since the p r o p e l l e r d i s c i s

g r e a t e r than the jet e x h a u s t a r e a , it follows t h a t f o r m i n i m u m

kinetic e n e r g y l o s s in the s l i p s t r e a m , ideally a l l work f r o m the

t u r b o p r o p cycle will be done by the p r o p e l l e r ,

E f f e c t s of S m a l l Changes in B a s i c Cycle F a r a m e t e r s ,

-

In o r d e r t o obtain a n e x p r e s s i o n f o r the change in t o t a l work out-

put due t o a s m a l l change

G

in the o p t i m u m jet p r e s s u r e r a t i o

X9&

, the t h i r d t e r m of the following T a y l o r expansion was calcu-

l a t e d :

where h a s been s e t e q u a l t o z e r o and defines the opti-

m u m jet p r e s s u r e r a t i o X*

.

F o r ideal conditions

(?+

=

Cv=

(

,

X%

&&-I)+

,)

,

t h i s ex-

K

- 1 1 -

The effect on t o t a l work output of t h e c y c l e of a s m a l l

change in

d

( r e l a t e d t o the p r e s s u r e r a t i o a c r o s s the com-

p r e s s o r ) was found t o be:

F o r I ,

&k!)+

1

,

t h i s r e d u c e s t o

K

The effect on t o t a l w o r k of a s m a l l change in K ( r e l a t e d t o t e m p - e r a t u r e r i s e in the combustion c h a m b e r ) i s :

- C

-

1 ,

=X+=

b + @ - l ) +

I

F o r

f p =

76

-

,,-

a t h i s r e d u c e s to:

K

The effect on t o t a l work of a s m a l l change i n ( r e l a t e d t o

Effect of S m a l l Changes in Component E f f i c i e n c i e s on the P o w e r

Coefficient. The effect of s m a l l changes in component efficiencies

on the optimum jet p r e s s u r e r a t i o X:g f o r the idealized cycle was

found by d e t e r m i n i n g the coefficients in the following T a y l o r ex-

pansion:

+-

ax+

(

)

+

a x *

+

..

- - - .

Jcl-74

I,,,,

as-cv)

b,,,

T o obtain t h e s e c o e f f i c i e n t s , the r e l a t i o n defining the optimum jet

p r e s s u r e r a t i o X* was differentiated in r e s p e c t t o the various ef-

f i c i e n c i e s and evaluated a t the point of i d e a l efficiency. The fol-

lowing d e r i v a t i v e s w e r e obtained:

a x *

=.

and the r e s u l t i n g T a y l o r expansion i s :

The coefficients of t h i s expansion give the changes i n the optimum

jet p r e s s u r e r a t i o X* f r o m the i d e a l value when the component ef-

f i c i e n c i e s a r e v a r i e d s l i g h t l y f r o m 100%. The coefficients a l s o

show the r e l a t i v e e f f e c t s on X'k of the v a r i o u s component e f f i c i e n c i e s of the c y c l e .

The effect on t o t a l work was obtained by a s i m i l a r p r o c e d u r e .

The e x p r e s s i o n f o r the optimum jet p r e s s u r e r a t i o We was i n s e r t e d

in the t o t a l work e q u a t i o n , and the d e r i v a t i v e s of t o t a l work in r e -

s p e c t t o the component e f f i c i e n c i e s was obtained. In t h i s c a s e , X*

i s a function of the e f f i c i e n c i e s a l s o , and the d e r i v a t i v e s obtained

above w e r e i n s e r t e d a p p r o p r i a t e l y

.

The following d e r i v a t i v e s w e r e

And a s i m i l a r T a y l o r expansion f o r

Cp

,

the power coefficient, is:

Effect of S m a l l Changes in Component E f f i c i e n c i e s on the

Specific F u e l Consumption. The effect of component e f f i c i e n c i e s

on the s p e c i f i c f u e l consumption S was obtained a s follows:

3600 f .lb f u e l

With the p r o p e r e x p r e s s i o n s f o r f u e l consumption f and t o t a l work in t e r m s of the power coefficient

Cp

,

t h i s b e c o m e s :

If the power coefficient

Cp

in the p r e c e d i n g T a y l o r expansion i s c o n s i d e r e d t o be e q u a l t o

Cp,

-

OC

,

w h e r e

Clo

i s the power

coefficient with a l l e f f i c i e n c i e s 100O/~ and with optimum jet p r e s s u r e

f o r m by neglecting s q u a r e s of s m a l l q u a n t i t i e s . The expansion

thus obtained f o r the s p e c i f i c f u e l consumption i s :

H e r e

So=

and i s the value of the s p e c i f i c f u e l

consumption under i d e a l conditions

.

By evaluating the coefficients

of t h i s expansion f o r any given s e t of cycle p a r a m e t e r s , the r e l a -

tive effect of the component e f f i c i e n c i e s on the s p e c i f i c f u e l con-

-16

-

11. ANALYSIS O F CYCLE WITH REGENERATION

A s c h e m a t i c d i a g r a m of the t u r b i n e - p r o p e l l e r engine with r e g e n e r a t i o n i s given a s F i g u r e 2. It i s a s s u m e d t h a t a n i d e a l h e a t exchanger is i n s t a l l e d a f t e r the c o m p r e s s o r and a f t e r the

t u r b i n e s o that a s m a l l amount of h e a t c a n be t a k e n f r o m the ex-

h a u s t g a s e s t o h e a t the a i r leaving the c o m p r e s s o r . The s a m e

a s s u m p t i o n s and s y m b o l s a s f o r the b a s i c cycle a r e used in the

de ve lopme nt of the equations

.

C o m p r e s s i o n in the d i f f u s e r :

C o m p r e s s i o n in the c o m p r e s s o r :

Work of non-ideal c o m p r e s s i o n :

T o t a l h e a t added in the combustion c h a m b e r (including h e a t f r o m

Setting the work of the t u r b i n e (to s t a t i o n 4) e q u a l t o the work of the c o m p r e s s o r , the following r e l a t i o n s a r e obtained:

R e g e n e r a t o r heating e f f e c t i v e n e s s is defined a s :

P o w e r t o d r i v e the p r o p e l l e r :

A s s u m i n g no p r e s s u r e l o s s a c r o s s t h e r e g e n e r a t o r , the jet p r e s -

s u r e r a t i o s f o r the cycle with and without r e g e n e r a t i o n a r e the

The t e m p e r a t u r e r a t i o s and

2

a r e found t o be:

7;;

-7z

With t h e s e t e m p e r a t u r e r a t i o s the t h r u s t velocity

G''

c a n be ob-

tained:

The t o t a l w o r k output of the t u r b o p r o p cycle with r e g e n e r a t i o n i s :

F u e l Consumption D e c r e a s e due t o R e g e n e r a t i o n . F o r the

b a s i c cycle without r e g e n e r a t i o n , the a c t u a l f u e l consumption in

lb f u e l

was found t o be: s e c

/6

f u e l

A s s u m i n g the s a m e t e m p e r a t u r e r i s e in the combustion c h a m b e r

f o r the cycle with and without r e g e n e r a t i o n , the a c t u a l f u e l con-

sumption with r e g e n e r a t i o n i s found a s follows:

f r o m which f " c a n be d e t e r m i n e d a s :

T h u s , the a c t u a l f u e l consumption d e c r e a s e due t o r e g e n e r a t i o n :

P o w e r Output D e c r e a s e due t o R e g e n e r a t i o n . The c o r r e s -

the s a m e t e m p e r a t u r e r i s e in the combustion c h a m b e r ) b e c o m e s :

Optimum J e t P r e s s u r e R a t i o with R e g e n e r a t i o n , By dif-

f e r e n t i a t i n g the t o t a l work e x p r e s s i o n f o r the r e g e n e r a t i v e cycle

with r e s p e c t t o X and s e t t i n g the d e r i v a t i v e e q u a l t o z e r o , the e x - p r e s s i o n d e t e r m i n i n g the optimum jet p r e s s u r e r a t i o i s d e t e r m i n e d

t o be:

Effect of S m a l l Changes in the Component E f f i c i e n c i e s of

the R e g e n e r a t i v e T u r b o p r o p Cycle on Optimum J e t P r e s s u r e R a t i o ,

P o w e r Coefficient and Specific F u e l Consumption, S i m i l a r T a y l o r

expansions w e r e c a r r i e d out f o r the t u r b o p r o p cycle with r e g e n e r -

a t i o n t o obtain e x p r e s s i o n s f o r optimum jet p r e s s u r e r a t i o , t o t a l

w o r k , and s p e c i f i c f u e l consumption f o r s m a l l v a r i a t i o n s in

7*

7 p 9

7~

Cv

and

qX

.

Because of the f a c t t h a t the coefficients

a r e evaluated a t the point of i d e a l e f f i c i e n c y , the d e r i v a t i v e s of X%

and

Wtrt

in r e s p e c t t o

tp

,

,

7 c

and

Cv

r e m a i n un-

and the complete expansions f o r X* and

Week

about the point of i d e a l e f f i c i e n c i e s a r e :

The coefficients of t h e s e expansions a r e plotted i n F i g u r e s 3

-

9

f o r v a r i o u s values of

_{P ?}

b-,

and K , and show the r e l a t i v e e f f e c t s on optimum jet p r e s s u r e r a t i o and t o t a l work output of s m a l l changes

in t h e component e f f i c i e n c i e s f r o m the i d e a l value.

An expansion f o r s p e c i f i c f u e l consumption S of the r e g e n -

e r a t i v e cycle was obtained by a method s i m i l a r t o t h a t d e s c r i b e d

p r e v i o u s l y . By dropping t e r m s s u c h a s

Tx

((I-(?&) and

qx(,-qc)

Relative values of the coefficients of t h i s expansion a r e plotted on

F i g u r e 10 f o r

Mo

= . 5 0 ,

fi

= 3 and v a r i o u s values of K t o show

Pl

the e f f e c t s of component efficiencies on the s p e c i f i c f u e l consump-

[image:27.507.39.479.76.269.2]

111, DISCUSSION O F RESULTS

By the technique d e s c r i b e d in the f o r e g o i n g s e c t i o n s , i t

i s possible t o d e t e r m i n e the r e l a t i v e e f f e c t s o n p e r f o r m a n c e c r i -

t e r i a of a s m a l l change in any of the component e f f i c i e n c i e s of

the cycle f r o m t h e i r i d e a l v a l u e s , The e x p a n s i o n s s u b m i t t e d

a s Equations ( 7 6 ) , (7'7), and (78) give t h e s e v a r i a t i o n s a s func- t i o n s of only t h r e e of the cycle p a r a m e t e r s

d Z p ,

and K. T h i s s i m p l i f i c a t i o n was m a d e p o s s i b l e by the introduction of the op-

t i m u m jet p r e s s u r e r a t i o X*. The value of X* f o r t h e i d e a l cycle

0

( a l l component e f f i c i e n c i e s 100

/o)

w a s found t o be a n explicit

function of

T a p ,

and K , and had a value s u c h t h a t a l l of the work output of the cycle went t o d r i v e t h e p r o p e l l e r , none going

t o jet t h r u s t .

The expansion (Equation 76) f o r the optimum jet p r e s -

-

24-

A s the e f f i c i e n c i e s of the t u r b i n e and p r o p e l l e r d e c r e a s e f r o m the i d e a l v a l u e , the value of the o p t i m u m jet p r e s s u r e r a t i o i s

found t o i n c r e a s e , showing a n i n c r e a s e in power t o jet t h r u s t .

The effect of a change in

qt

o r

t?p

on X* i s of the s a m e o r d e r , the t u r b i n e efficiency having the g r e a t e r value. The effect of

a d e c r e a s e in the jet e x h a u s t nozzle coefficient

Cy

o r a n in-

c r e a s e in r e g e n e r a t o r e f f e c t i v e n e s s

qx

i s shown t o lower the

value of X * . C o m p r e s s o r efficiency

/k

i s found t o have no ef-

f e c t , The r e l a t i v e i m p o r t a n c e of changes i n component elfic- i e n c i e s on the value of the optimum jet p r e s s u r e r a t i o X* is shown in F i g u r e s 3-61? P a r t

VII,

f o r v a r i o u s values of

d

_?/U2

and K ,

A s i m p l e e x p r e s s i o n c a n be d e r i v e d f r o m S q u a t i o n s

( 7 6 )

and ( 7 7 ) t o show how the r a t i o of jet work t o p r o p e l l e r work

v a r i e s a s the value of jet p r e s s u r e r a t i o v a r i e s f r o m the opti-

m u m . A s s u m i n g , a s a f i r s t a p p r o x i m a t i o n , t h a t

(X

- X I )

<<

I

the following r a t i o is obtained:

Jet

Work p r o p e l l e r Work

The coefficient of (X-x*) gives the slope of the s t r a i g h t line showing the above r a t i o . F o r

fl.

=

.50, = 3 , and

K

= 4 ,

P

-

25

-

The power coefficient expansion

Cp

(Equation (77)

)

a s

a function of the cycle p a r a m e t e r s and e f f i c i e n c i e s was found t o be:

A d e c r e a s e in the component e f f i c i e n c i e s

_7t.

_{7 p *}

and f r o m t h e i r i d e a l values i s shown t o r e s u l t in a c o r r e s p o n d -

ing d e c r e a s e in

cp

and hence in the power output of the c y c l e .

The effect of the t u r b i n e efficiency i s p r e d o m i n a n t i n d e c r e a s i n g

the power output, followed by p r o p e l l e r efficiency, c o m p r e s s o r

e f f i c i e n c y , and jet e x h a u s t nozzle coefficient in r e l a t i v e i m p o r t a n c e .

R e g e n e r a t i o n was found t o d e c r e a s e the power output of the i d e a l

t u r b o p r o p c y c l e . F i g u r e s

7-9,

P a r t V I I , show the r e l a t i v e i m - p o r t a n c e of t h e s e coefficients f o r v a r i o u s values of the cycle pa-

r a m e t e r s . F o r t y p i c a l cycle p a r a m e t e r values of

a=

.50,&==3

f t

and K = 4, a lo/o change in

7 !

r e s u l t s in a change in

C

of

-

.01146

,

P

while a 1% change in % r e s u l t s in a change of -.00510 in the value

change in c o m p r e s s o r efficiency i s independent of the amount of

h e a t added in the combustion c h a m b e r and depends on flight s p e e d

and c o m p r e s s o r r a t i o only.

T o show the v a r i a t i o n of

Cp

with change in t u r b i n e and

c o m p r e s s o r e f f i c i e n c y , values of power coefficient a r e plotted

a g a i n s t c o m p r e s s o r p r e s s u r e r a t i o on F i g u r e 11 f o r

_qx

_{= 0 ,}

qp=

_1,

CV=

1,

/Ye=

* 5 0 and K = 4. The e x p a n s i o n given in Equation (77) was used f o r the plot which shows how the c o m p r e s s o r p r e s - s u r e r a t i o f o r m a x i m u m power coefficient change s with the effic-

i e n c i e s of c o m p r e s s o r and t u r b i n e .

Specific f u e l consumption S a s a function of the cycle pa-

r a m e t e r s and component e f f i c i e n c i e s i s given a s Equation (78). The coefficients of t h i s expansion a r e plotted on F i g u r e I2 f o r a

t y p i c a l s e t of values of

S,,U,

and K . The t u r b i n e and p r o p e l l e r e f f i c i e n c i e s a r e s e e n t o be predominant in i n c r e a s i n g t h e f u e l con-

s u m p t i o n , L e s s e r e f f e c t s a r e shown f o r c o m p r e s s o r efficiency

and e x h a u s t nozzle coefficient, F o r cycle p a r a m e t e r values of

Me=

.5 0 ,

fi

= 3 , and K =

4 ,

a 1°/o change in t u r b i n e efficiency

PI

I& r e s u l t s in a n i n c r e a s e of the s p e c i f i c f u e l consumption of

0

.0061 (Ib f u e l / h p . h r

.)

while a l /o change in c o m p r e s s o r efficiency

7,

r e s u l t s i n a n i n c r e a s e of .0015 (lb.fuel/hp.hr

.).

T h e s e

t y p i c a l values show the r e l a t i v e effect of t u r b i n e efficiency a s

-27-

t i o n of the i d e a l t u r b o p r o p c y c l e .

R e g e n e r a t i o n , a s c o n s i d e r e d i n t h i s a n a l y s i s , d e c r e a s e s

the power coefficient and d e c r e a s e s the s p e c i f i c f u e l consumption.

0

F o r

Vx

=

l o

/o, and f o r the p a r a m e t e r values given above, i t i s

found t h a t

cp

is d e c r e a s e d by

-

.00246 while S i s d e c r e a s e d by .I78 (lb fuel/hp,hr

.)

.

The r e s u l t s of t h e s e calculations c a n be used analytically

t o i m m e d i a t e l y show the effect and i m p o r t a n c e of the e f f i c i e n c i e s

i n a t u r b i n e - p r o p e l l e r engine

.

The r e l a t i v e i m p o r t a n c e of the ef-

f e c t s of t h e s e e f f i c i e n c i e s on p e r f o r m a n c e c r i t e r i a c a n be e a s i l y

deduced f r o m s i m p l e m a t h e m a t i c a l e x p r e s s i o n s without the use of

complex c h a r t s and g r a p h s . As a n e x a m p l e , by the u s e of Equa-

tion (77) i t i s p o s s i b l e t o show how the p e r c e n t a g e work done by

the p r o p e l l e r changes with v a r i a t i o n s in component e f f i c i e n c i e s .

F o r the i d e a l cycle with o p t i m u m jet p r e s s u r e r a t i o X*

,

i t was

found t h a t the t o t a l power output of the c y c l e went t o p r o p e l l e r work.

A s t h e component e f f i c i e n c i e s c h a n g e , t h e power division between jet and p r o p e l l e r likewise c h a n g e s .

T h i s type of cycle a n a l y s i s c a n be used t o solve v a r i o u s

t y p e s of p r o b l e m s . As a n e x a m p l e , Equation (29) c a n be used t o

d e t e r m i n e the value of c o m p r e s s o r p r e s s u r e r a t i o t h a t will give

the m a x i m u m power output of the cycle f o r any given values of

-

28-

in Equation (28) and introducing the value of the optimum jet p r e s s u r e r a t i o , the following e x p r e s s i o n i s obtained:

T h i s d e r i v a t i v e is s e t e q u a l t o z e r o toifind the value of which

gives m a x i m u m

C

f o r the specified p a r a m e t e r values:

P

d=

C j X

rU-

(for m a x .

c~

1

T h i s s i m p l e equation gives the value of c o m p r e s s o r p r e s s u r e

r a t i o f o r m a x i m u m power output and c a n be plotted f o r v a r i o u s

values of c o m p r e s s o r efficiency t o show the t r e n d with change in

efficiency.

S i m i l a r p r o b l e m s c a n be Likewise e a s i l y s o l v e d using

Equations (30) and (31) t o give the values of/ and

M

f o r optimum

- 2 9 -

IV. CONCLUSIONS

F r o m the r e s u l t s of t h i s cycle a n a l y s i s of the g a s t u r b i n e

p r o p e l l e r - d r i v i n g engine,

it

i s concluded that:

1 . A complete a n a l y s i s of the p e r f o r m a n c e of the i d e a l t u r b o p r o p cycle c a n be m a d e in t e r m s of t h r e e b a s i c p a r a m e t e r s

b

W,,, and K by introducing the o p t i m u m jet p r e s s u r e r a t i o X*

a s a m e a n s of dividing the power output between p r o p e l l e r and

jet t h r u s t .

2.

An a n a l y s i s of the non i d e a l t u r b o p r o p c y c l e , with o r

without r e g e n e r a t i o n , c a n be m a d e in t e r m s of the f o u r p a r a m e -

t e r s ,

6,y,

K and X and the component e f f i c i e n c i e s of the s y s t e m . The r e l a t i v e e f f e c t of t h e s e e f f i c i e n c i e s on p e r f o r m a n c e c r i t e r i a

c a n be shown by a T a y l o r expansion about the point of i d e a l effic-

i e n c i e s . In t h i s m e t h o d , the optimum jet p r e s s u r e r a t i o X2k c a n

a g a i n be i n t r o d u c e d , and the r e s u l t i n g p e r f o r m a n c e e x p r e s s i o n s

c a n be given i n t e r m s of t h r e e b a s i c p a r a m e t e r s

d,,U,

and

K .

3 , Of the component e f f i c i e n c i e s , the t u r b i n e efficiency

Vt

was found t o be p r e d o m i n a n t in affecting the p e r f o r m a n c e c r i -

t e r i a of the i d e a l t u r b o p r o p c y c l e .

4.

P r o p e l l e r efficiency was a l s o found t o have a n i m -

' l p

portant effect on the p e r f o r m a n c e of the i d e a l cycle s i n c e , f o r

- 3 0 -

5 . R e g e n e r a t i o n , a s c o n s i d e r e d in the a n a l y s i s , h a s the effect of d e c r e a s i n g the t o t a l power output and d e c r e a s i n g the

[image:36.544.15.526.65.677.2]

3L

SCHEMATIC DIAGRAMS OF

TURBOPROP CYCLES

FIG. I

BASIC TURBOPROP CYCLE

F I G .

2

VI. APPENDIX A

SYMBOLS AND DEFINITIONS

C

-

Specific h e a t a t constant p r e s s u r e

[EF).

P

W t o t

-

P o w e r coefficient, defined a s

-

C P t d '

-

P o w e r coefficient with a l l e f f i c i e n c i e s 100°/0 and optimum

jet p r e s s u r e r a t i o .

C <, _Y

-

e x h a u s t nozzle velocity c o e f f i c i e n t .

( r e l a t e d t o p r e s s u r e r i s e a c r o s s the c o r n p r e s s o r ) .

E

-

A s m a l l change in a given p a r a m e t e r .

I c

-

Efficiency of c o m p r e s s o r .

8

-

P r o p e l l e r e f f i c i e n c y , including g e a r box l o s s e s and other power t r a n s m i s s i o n l o s s e s .

'7t

-

T u r b i n e efficiency.

3 x

-

R e g e n e r a t o r heating e f f e c t i v e n e s s .

-

Actual f u e l consumption

(

l b f u e l

s e c

1

2 g

-

A c c e l e r a t i o n due t o g r a v i t y (32.17 f t / s e c

Y

-

R a t i o of s p e c i f i c h e a t s

H

-

Heating value of the f u e l

(

BTU

Ib fue 1

)

J

-

Mechanical equivalent of h e a t (778.3

-)

f t l b

BTU

K

- -

-

r3 ( r e l a t e d t o the t e m p e r a t u r e r i s e in the combustion

To

c h a m b e r

M

-

L o c a l Mach number

-

-

-

" ( r e l a t e d t o f o r w a r d flight s p e e d )

To

lb

-

S t a t i c p r e s s u r e

(-)

in 2

BTU

-

Heat added p e r pound of a i r

(-)

l b

-

Specific f u e l consumption

(

l b f u e l

+-hour

)

-

Specific f u e l consumption with a l l e f f i c i e n c i e s e q u a l t o

0

100 /o and with optimum jet p r e s s u r e r a t i o

0

-

F r e e s t r e a m s t a t i c t e m p e r a t u r e

(

R)

-

T o t a l t e m p e r a t u r e (OR)

-

T o t a l t e m p e r a t u r e f o r a n i d e a l i s e n t r o p i c p r o c e s s

-

T o t a l t e m p e r a t u r e a t c e r t a i n points i n the r e g e n e r a t i v e cycle

-

Velocity (ft/sec)

BTU

-

Work p e r pound of a i r

(-)

as'

l b

( r e l a t e d t o the p r e s s u r e r a t i o a c r o s s

the e x h a u s t nozzle)

-

Optimum jet p r e s s u r e r a t i o f o r i d e a l division of power

P A R T VII

On the following F i g u r e s

3-6

a r e plotted the coefficients

of the expansion of optimum jet p r e s s u r e r a t i o X* a s a function of the cycle p a r a m e t e r s

b,

_{P S}

and K , and the component ef- f i c i e n c i e s $+a

Cvs

7 c

and

? X .

Equation (76):

The following notation h a s been used in plotting:

(ideal o p t i m u m jet p r e s s u r e r a t i o )

(coef. of (I-

p)

t e r m )

(coef. of (1- p) t e r m )

K

7

zy

=

4 3 (coef. of (1-c,,) t e r m )

Zs=

0 (coef. of (1

-p)

t e r m )

On the following F i g u r e s 7-9 a r e plotted the coefficients o f

the expansion of the power coefficient

C2

a s a function of the cycle p a r a m e t e r s

$

,U,, and K and the component efficiencies

p1

+'PI

Cv,

qk

and

q x

.

Equation (77)

The following notation h a s b e e n used in plotting:

=

(

)

- I (ideal power coefficient)

K %

* v =

- [ K + h ( N - o

--I

(coef. of (1

-

p)

t e r m )

REFERENCES

1 , NACA TN 2254, J a n u a r y 1951, " R e g e n e r a t o r - D e s i g n Study and Its Application t o T u r b i n e - P r o p e l l e r Engines"

-

S . V.Manson 2. NACA TN 2178, S e p t e m b e r 1950, "Method of D e t e r m i n i n g Opti-

m u m Division of P o w e r Between J e t and P r o p e l l e r f o r Max- i m u m T h r u s t P o w e r of a T u r b i n e - P r o p e l l e r Engine", A.M. T r o u t and E .

W .

Hall

3. NACA TN 1497, J a n u a r y 1948, "A Method of Cycle Analysis f o r A.ir c r a f t Gas- Turbine P o w e r P l a n t s Driving P r o p e l l e r s l l

,

It.

E.

E n g l i s h and C

.

V . Wauser.

4 . NACA TN 176 2 , D e c e m b e r 1948, "Control C o n s i d e r a t i o n s f o r Optimum-Power P r o p o r t i o n m e n t in Turbine- P r o p e l l e r En- g i n e s " , M. F

.

Heidmann and D, Novik

5 .

D a v i s , S , J . , and F a w z e , M . I.: "The E f f i c i e n c i e s of Combus- tion T u r b i n e s " , Engineering, vole 157, Nos, 4078 and 4080, M a r c h 10 and 24, 1944, pp, 401-403, 421-424.

6 .

S a l i s b u r y , J .

M.:

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